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The vigesimal or base- numeral system is based on twenty (in the same way in which the ordinary decimal numeral system is based on ten). Twenty is the sum of all fingers and toes on unmutated human hands and feet, and is the product of five and four.
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Places
In a vigesimal place system, twenty individual numerals (or digit symbols) are used, ten more than in the usual decimal system. One modern method of finding the extra needed symbols is to write ten as the letter A20 (the 20 means base 20), to write nineteen as J20, and the numbers between with the corresponding letters of the alphabet. (This is an extension of the common computer-science practice of writing hexadecimal numerals over 9 with the letters "A-F".) Another method skips over the letter "I", in order to avoid confusion between I20 as eighteen and 1 (one), so that the number eighteen is written as J20, and nineteen is written as K20. The number twenty is written as 1020.
According to this notation:
2020 means forty in decimal
DA20 means two hundred seventy in decimal
10020 means four hundred in decimal .
In the rest of this article below, numbers are expressed in decimal notation, unless specified otherwise. For example, 10 means ten, 20 means twenty.
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Usage
In many languages, especially in Europe, 20 is a base, at least with respect to the linguistic structure of the names of certain numbers (though a thoroughgoing consistent vigesimal system, based on the powers 20, 400, 8000 etc., is not generally used).
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In Europe
According to German linguist Theo Vennemann, the vigesimal system in Europe is of Basque (Vasconic) origin and spread from the so-called Vasconic languages to other European tongues, such as many Celtic languages, French and Danish.
According to Menninger, the vigesimal system originated with the Normans and spread through them to Western Europe, the evidence being that Celtic languages often use vigesimal counting systems. Others believe that this theory is unlikely, however.
Twenty () is used as a base number in the French language names of numbers from 60 to 99. So for example, means 4 times 20, i.e. is the French word for 80, and (literally "sixty-fifteen") means 75.
Twenty () is used as a base number in the Danish language names of numbers from 50 to 99. For example, (short for ) means 3 times 20, i.e. 60. For details, see Danish numerals.
Twenty () is used as a base number in the Breton language names of numbers from 40 to 49 and from 60 to 99. For example, means 2 times 20, i.e. 40, and (literally "three-six and four-twenty") means 3×6 + 4×20, i.e. 98. However, 30 is and not ("ten and twenty"), and 50 is ("half-hundred").
Twenty () is used as a base number in the Welsh language, although in the latter part of the twentieth century a decimal counting system has come to be preferred (particularly in the South), with the vigesimal system becoming 'traditional' and more popular in North Welsh. means 2 times 20 i.e. 40, means 3 times 20 i.e. 60. Prior to the currency decimalisation in 1971, (6 times 20 paper) was the nickname for the 10 shilling (= 120 pence) note.
Twenty () is used in an older counting system in Irish Gaelic, though most people nowadays use a decimal system, and this is what is taught in schools. Thirty is , literally twenty and ten. Forty is , literally two twenties. is sixty (three twenties) and is eighty (literally four twenties).
Twenty () is used as a base number in the Albanian language. The word for 40 () means two times 20.
Twenty () is used as a base number in the Basque language for numbers up to 100 (). The words for 40 (), 60 () and 80 () mean "two-score", "three-score" and "four-score", respectively. The number 75 is called , lit. "three-score-and ten-five". The Basque nationalist Sabino Arana proposed three vigesimal digit systems to match the spoken language but they are mostly forgotten.
In English, counting by the score has been used historically, as in the famous opening of the Gettysburg Address "Four score and seven years ago…", meaning eighty-seven (87) years ago. This method has fallen into disuse, however.
Twenty () is used as a base number in the Georgian language. For example, 31 () literally means, twenty-and-eleven. 67 () is said as, “three-twenty-and-seven”.
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Outside Europe
In East Asia, the Ainu language also uses a counting system that is based around the number 20. “” is 20, “” (ten more until two twenties) is 30, “” (two twenties) is 40, “” (five twenties) is 100. Subtraction is also heavily used, e.g. “” (one more until ten) is 9.
Twenty was a base in the Maya and Aztec number systems. The Maya used the following names for the powers of twenty: (20), (202 = 400), (203 = 8,000), (204 = 160,000), (205 = 3,200,000) and (206 = 64,000,000). See also Maya numerals and Maya calendar, Mayan languages, Yucatec. The Aztec called them: (1 × 20), (1 × 400), (1 × 8,000), (1 × 20 × 8,000 = 160,000), (1 × 400 × 8,000 = 3,200,000) and (1 × 20 × 400 × 8,000 = 64,000,000). Note that the prefix at the beginning means "one" (as in "one hundred" and "one thousand") and is replaced with the corresponding number to get the names of other multiples of the power. For example, (2) × (20) = (40), (2) × (400) = (800). Note also that the in (and ) and the in are grammatical noun suffixes that are appended only at the end of the word; thus , and compound together as (instead of
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Related observations
Among multiples of 10, 20 is described in a special way in some languages. For example, the Spanish words (30) and (40) consist of " (10 times)", " (10 times)", but the word (20) is not presently connected to any word meaning "two" (although, historically, it is). Similarly, in Semitic languages such as Arabic and Hebrew, the numbers 30, 40 ... 90 are expressed by morphologically plural forms of the words for the numbers 3, 4 ... 9, but the number 20 is expressed by a morphologically plural form of the word for 10.
In some languages, the names of the two-digit numbers from 11 to 19 consist of one word, but the names of the two-digit numbers from 21 on consist of two words. So for example, the English words eleven (11), twelve (12), thirteen (13) etc., as opposed to twenty-one (21), twenty-two (22), twenty-three (23), etc. In a number of other languages (such as Hebrew), the names of the numbers from 11-19 contain two words, but one of these words is a special "teen" form which is different from the ordinary form of the word for the number 10, and may in fact be only found in these names of the numbers 11-19.
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Further reading
Karl Menninger: Number words and number symbols: a cultural history of numbers; translated by Paul Broneer from the revised German edition. Cambridge, Mass.: M.I.T. Press, 1969 (also available in paperback: New York: Dover, 1992 ISBN 0-486-27096-3)
Levi Leonard Conant: The Number Concept: Its Origin and Development; New York, New York: MacMillon & Co, 1931. Project Gutenberg EBook
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Notes
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